Monument Valley is capable of taking your breath away one moment, and then breaking your brain in two the next. To traverse these impossible worlds you'll need to suspend your usual way of thinking altogether.

A full on Metroidvania adventure (a game with a sprawling, interconnected world, for those not au fait with the language) set inside a pyramid. Chockablock with secrets and fun upgrades, Traps is easy to fall in love with.

Square Enix has a whole host of these apps: smart turn-based puzzlers featuring heroes from console games. Hitman GO, which turns Agent 47's murderous shenanigans into a classy tabletop game, should run well on your new phone.

An interactive story with way more than 80 ways for it to turn out. You're constantly making choices that might have a tiny impact on the narrative, or might lead you to be kidnapped by pirates. Who knows?

Draw a squiggle on the screen. Watch it endlessly repeat itself as it wriggles across the screen. Now use this to score through coloured dots, and avoid the black ones. Every solution is utterly personal.

For some, it will be a headache simulator. Those who get into Super Hexagon's rhythm, though, will find a brutally brilliant arcade gem about spinning in circles and dodging danger. All to a catchy chiptune beat.

The first man on Mars won't be a super soldier with a flamethrower: it will be a scientist. Follow his journey in Waking Mars: an organic puzzle adventure about managing ecosystems (and uncovering secrets).

This game is all about iteration. You're trying to make a train system to deliver a bunch of carriages to the right station, but your first attempt will likely fail. Time to tweak, bug fix, and iterate until you get it right.